Problem: Factor the following expression: $-3$ $x^2$ $-7$ $x$ $-4$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-3)}{(-4)} &=& 12 \\ {a} + {b} &=& & & {-7} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $12$ and add them together. The factors that add up to ${-7}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-4}$ and ${b}$ is ${-3}$ $ \begin{eqnarray} {ab} &=& ({-4})({-3}) &=& 12 \\ {a} + {b} &=& {-4} + {-3} &=& -7 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-3}x^2 {-4}x {-3}x {-4} $ Group the terms so that there is a common factor in each group: $ ({-3}x^2 {-4}x) + ({-3}x {-4}) $ Factor out the common factors: $ x(-3x - 4) + 1(-3x - 4) $ Notice how $(-3x - 4)$ has become a common factor. Factor this out to find the answer. $(-3x - 4)(x + 1)$